Enhanced Summary of G.P. Agrawal Nonlinear Fiber Optics (3rd ed) Chapter 9 on SBS
Stimulated Brillouin scattering is a nonlinear three-wave interaction between a forward-going
laser pump beam P, a forward-going acoustic wave Q, and a backward-traveling Stokes beam S.
The pump field generates the acoustic wave via electrostriction. That is, the electric field of the
pump wave exerts oppositely directed forces on the positive and negative ions or domains in the
silica, inducing a strain which generates the sound wave. In turn, that acoustic field modulates
the refractive index, resulting in a grating that Bragg scatters the pump into the Stokes beam. The
Stokes scattered light is downshifted in frequency by the Doppler shift, because the grating is
moving forward at the acoustic (sound) speed, which for fused silica is A = 5960 m/s .
(Quantum mechanically, a pump photon is annihilated and a Stokes photon and an acoustic
phonon are simultaneously generated.) Energy conservation requires that the Brillouin frequency
shift be ( P S ) / 2 = B / 2 B , while momentum conservation says that the acoustic
wavevector must be k B = k P k S kB = kP + kS since the Stokes wave travels backward along
the optical fiber. But B = A kB = 2n A 2 / P because kS kP . Here the refractive index of
silica is n = 1.45 . Thus for a vacuum pump wavelength of P = 1.55 μm , the shift is
B = 2n A / P = 11 GHz . Acoustic waves are assumed to decay as exp( Bt) where the
acoustic phonon lifetime in silica is TB 1 / B 10 ns . The SBS gain spectrum is assumed to
be Lorentzian with angular frequency dependence
g B ( B / 2)2
(1)
( B )2 + ( B / 2)2
and thus its FWHM is B = B / 2 15 MHz . Here the peak Brillouin gain coefficient is
gB =
2
n7 p12
(2)
cP2 A B
where the vacuum speed of light is c = 3 108 m/s , the density of silica is = 2200 kg / m 3 ,
and the measured longitudinal elasto-optic coefficient of silica is p12 = 0.286 . [The LorentzLorenz relation predicts that p12 = (n2 1)(n2 + 2) / (3n4 ) = 0.34 in rough agreement.] In bulk
silica, values of B between 10 and 20 MHz have been measured at a pump wavelength of 1.5
μm; I will the split the difference and use B,bulk = 15 MHz in this review. In a fiber, the width
depends on the numerical aperture (NA) according to Kovalev & Harrison, Opt. Lett. 27:2022
(2002) as
2
NA ( B,fiber ) = ( B,bulk ) + B 2 n 2
4
2
because large numerical apertures imply that Stokes beams can travel at angles to the exact
backward direction, thus slightly relaxing the momentum conservation condition stipulated
1
(3)
above. Equation (3) is plotted in Fig. 1. Inhomogeneities in the fiber cross section can similarly
relax the wavevector condition and thereby increase the Brillouin bandwidth.
Fig. 1. Variation in the Brillouin gain FWHM with the numerical
aperture of a silica fiber at 1.5 μm.
Substituting the preceding values for bulk silica at 1.5 μm into Eq. (2) gives
g B 2 1011 m/W . Theory predicts that B B2 P2 , which implies that gB should be
independent of the pump wavelength. Note that if the polarization angle between the pump and
Stokes beams varies randomly (as in a non-polarization-maintaining fiber), then the value of gB
needs to be reduced by a factor of 1.5.
Steady-state conditions
Suppose that the pump is at least quasi-CW. Then the coupled intensity equations are
dI P
dz
= g B I P IS I P
(4)
for the pump as a function of length z along the fiber ranging from 0 to L, and
dIS
dz
= g B I P IS + IS
(5)
for the Stokes beam. If the attenuation coefficient (assumed to be the same at the pump and
Stokes wavelengths) is negligible, then 0 and it immediately follows from Eqs. (4) and (5)
that I P IS = C , some constant intensity difference.
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Brillouin threshold in the absence of losses
If we neglect both pump depletion and losses (the latter implying 0 ), then IP can be
taken to be constant along the length L of the fiber, and Eq. (5) can be separated and integrated to
get
IS (0)
IS ( L)
dIS
IS
0
= g B I P dz IS (0) = IS (L)exp(g B PL / A)
(6)
L
where P = I P A is the (input) pump power and A is the (modal) core area of the fiber.
Equation (6) applies to Brillouin amplification whereby a Stokes signal is input at z = L . If
instead the Stokes beam grows from noise, we model it by injecting one (fictitious) photon per
mode at a distance where the gain and loss balance; the method of steepest descent can then be
used to show that the critical pump power Pc defining the Brillouin threshold is
g B Pc L / A 21 .
(7)
Assuming telecom fiber values of A = 25 μm 2 , L = 25 km , and g B = 2 1011 m/W then
Eq. (7) implies Pc = 1 mW . It is precisely because this threshold is so low that SBS is significant
in fibers.
Gain saturation in the absence of losses
Now turn on pump depletion but continue to neglect losses, so that I P = IS + C . Then Eq. (4)
becomes
IS ( z)
z
dIS 1
1 C I I + C = gB dz S
S
(0)
0
IS
IS (z)
IS (0)
=
IS (z) + C
IS (0) + C
exp(g BCz) .
(8)
Replacing C by I P (0) IS (0) , Eq. (8) rearranges into
IS (z) =
b0 (1 b0 )
G(z) b0
I P (0)
(9)
where G(z) = exp[(1 b0 )g0 z] with
b0 IS (0)
I P (0)
and g0 g B I P (0) .
(10)
Here b0 measures the SBS reflectivity (fraction of the input pump power returned as output
Stokes power) and g0 is the small-signal SBS gain (in units of inverse distance).
Defining the ratio of input signals as bin IS (L) / I P (0) , the pump and Stokes powers
(normalized to the input pump power) are plotted in Fig. 2 for bin = 0.5% and g0 = 10 / L . For
this purpose, one needs to compute the value of b0. Evaluating Eq. (9) at z = L and solving for
b0 in the exponential gives
3
2
b (1 + bin ) b0
1
b0 = 1 ln 0
g0 L
bin
(11)
or b0 = 1 0.1ln(201b0 200b02 ) for our values of bin and g0. Iterating this formula starting from
b0 = 0.5 , it quickly settles down to b0 = 0.61273 , i.e., 61% of the input pump is converted into
output Stokes power, and thus C / I P (0) = 1 b0 = 0.38727 . Notice from the graph that most of
the power transfer occurs in the first quarter of the fiber’s length.
Fig. 2. Lossless intensities of the Stokes and pump beams as a
function of distance along the fiber’s length in dimensionless form.
The value g0 L = 10 used here corresponds to an unsaturated gain of Gu = exp(g0 L) = 22000 or
10 log(22000) = 43 dB , although the actual saturated gain is only
Gs IS (0) / IS (L) = b0 / bin = 120 due to pump depletion. Equation (9) evaluated at z = L
implies
bin =
b0 (1 b0 )
exp[(1 b0 )g0 L] b0
.
(12)
If b0 is small then bin b0 exp(g0 L) is also small and Gs Gu . But for larger values of bin the
gain rolls off, falling to about half at a saturation input Stokes power comparable to the threshold
pump power of about 1 mW.
To find the threshold exactly, suppose the Stokes input (seed) intensity is fixed, say at a
normalized value of g B IS (L)L = 105 k . Then g0 L = g B I P (0)L = k / bin where we now treat
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I P (0) —and hence bin—as a variable rather than as a constant, and plot the Stokes reflectivity b0
against it. For this purpose, define u 1 / bin and rewrite Eq. (12) as
u=
exp[k(1 b0 )u] b0
b0 (1 b0 )
.
(13)
A numerical root-finder in MatLAB was used to solve this equation for b0 as a function of u for
fixed k = 105 . The results are plotted in Fig. 3 where ku is on the horizontal axis, i.e., the
normalized pump intensity g B I P (0)L , and b0 is on the vertical axis, i.e., the efficiency or
“Stokes reflectivity” equal to the ratio of the Stokes output intensity to the pump input intensity.
The efficiency saturates at value 1 in the limit of very strong pumping. If we define the pump
threshold value to occur at an efficiency of 1 / e = 0.3679 , we see from the graph that it occurs at
a value of g0 L = 21 , in agreement with Eq. (7).
Fig. 3. Stokes efficiency as a function of the pump intensity in
dimensionless form.
Although it is not evident on the scale of the above plot, the curve actually turns around for
very low pump intensities and the efficiency then grows without bound. The reason for this
behavior is that for very weak pumping, the (fixed) Stokes input simply propagates straight
through the fiber and equals the Stokes output. Thus the efficiency becomes a constant divided
by the pump, which diverges in the limit of vanishing pump! This behavior becomes obvious if
one starts with a stronger seed, such as k = 0.1 . For that value, in fact, the turn around occurs
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near threshold, so that the efficiency merely shows a dip rather than a zero plateau at low pump
intensities.
Experimental results of Cotter in Electronics Letters 18:495 (1982)
An isolated Nd:YAG cw laser beam at 1.32 μm with a linewidth much smaller than the
Brillouin bandwidth B was sent into a 13.6 km fiber having losses of only 0.41 dB/km [so
that = 0.41ln(10) / 10 = 0.094 km 1 ], corresponding to an effective fiber length of
L
e
z
dz =
0
1 exp( L)
= 7.66 km .
(14)
(Note that the effective length equals L for a lossless fiber, but it decreases to 1 / if
>> 1 / L .) The effective fiber area was A = 47 μm 2 . At low input powers, the back-reflected
signal was 4% due to the air-fiber interface. The Brillouin threshold was reached at about 5 mW,
manifested as a substantial increase in backward-going power. Simultaneously, the transmitted
power dropped, saturating at about 2 mW for inputs exceeding 10 mW, corresponding to an SBS
conversion efficiency of about 65%. The Brillouin frequency shift was measured using a FabryPerot to be 12.7 GHz.
Time-dependent amplitude equations
Neglecting group velocity dispersion, as well as self and cross phase modulation, the coupled
equations for the complex pump and Stokes amplitudes AP and As are
AP
z
+
n AP 1
+ AP = g B ASQ
c t
2
2
(15)
and
AS
z
+
n AS 1
+ AS = g B AP Q *
c t
2
2
(16)
where Q is the acoustic power density (in W/m2) resulting from density variations of the silica
due to the sound wave, described by
TB
Q
+ Q = AP AS* .
t
(17)
For pump pulses of temporal width TP >> TB 10 ns , we can neglect Q / t . Define I P = AP
and IS = AS 2 , substitute Q = AP AS* into Eq. (15), multiply that equation through by AP* , and
add the result to its complex conjugate to get
I P
z
+
n I P
= g B I P IS I P
c t
2
(18)
which reduces to Eq. (4) under steady-state conditions. In a similar way, Eq. (16) becomes
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IS
z
+
n IS
= g B I P IS IS .
c t
(19)
Equations (18) and (19) exhibit relaxation oscillations with a period of twice the fiber transit
time, Tr = nL / c . (Physically, the Stokes power rapidly grows near the input end of the fiber,
thereby depleting the pump. That reduces the gain until the depleted portion of the pump exits
the far end of the fiber. The gain then builds back up and the process repeats, resulting in
oscillations.) In the presence of external feedback (for example due to back-reflections into the
fiber), these relaxation oscillations can develop into stable oscillations via self-induced intensity
modulation.
The graphs below show an example of relaxation oscillations in the absence of feedback, i.e.,
the pump and Stokes beams pass into and completely out of the fiber starting from opposite ends.
These were computed numerically using the Method of Lines (MOL) by modifying the MatLAB
code available online at http://www.scholarpedia.org/article/Method_of_lines. [Another
technique is to decouple the PDEs by defining u z + ct / n and w z ct / n so that Eq. (18)
becomes I P u = 2g B I P IS 2 I P and (19) becomes IS w = 2g B I P IS + 2 IS . This is
called the Method of Characteristics, but it has the disadvantage of mixing together the initial
conditions and boundary values.] It is convenient to rewrite Eqs. (18) and (19) in normalized
form by multiplying every term in them by L / I P (0) to get
P 1 P
+
= gPS P
z N t
(20)
S 1 S
+
= gPS S
z N t
(21)
and
where P I P (z) / I P (0) , S IS (z) / I P (0) , z z / L , t t / (N Tr ) , g g B I P (0)L , and
L . Here z and t both numerically evolve on grids running from 0 to 1, and N is the
number of transit times over which the simulation is permitted to run. In Fig. 4, I used a
normalized gain of g = 10 , normalized absorption of = 0.15 , N = 10 periods, 800 grid points
in the z direction, and 20 000 grid points in the t direction. (I found it is important to use finer
time than spatial discretization, so that the oscillations have time to settle down.) The spatial
derivatives are computed using five-point, fourth-order finite-difference approximations in
subroutine “dss004.m” (where some “for” loops were replaced with “parfor” to take advantage
of modern dual-core computer processing). The time integration is then performed using
subroutine “ode23” which implements the Bogacki-Shampine four-stage third-order adaptive
Runge-Kutta method. (Higher order methods were found to be too unstable.) The relative and
absolute tolerances were both set to 108 . The Stokes input relative to the pump input, bin, was
chosen to be 1%. To increase stability, I used the lossless steady-state profiles (corresponding to
b0 = 0.69064 ) for the pump and Stokes beams (similar to the curves in Fig. 2) as initial
conditions. Due to the loss, the normalized Stokes signal (left-hand graph) evolves away from its
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initial output value (at the front face z = 0 of the fiber) of b0 to a final value of 0.62548.
Meanwhile, the normalized pump signal (at the rear face z = 1) evolves from an initial value of
1 b0 + bin = 0.31936 to a final value of 0.30465.
Fig. 4. Time evolution of the output Stokes and pump beams,
showing relaxation oscillations over the course of 10 single-pass
transits through the fiber. The horizontal axis is in units of t / Tr .
If optical communication pulses of “0” and “1” are sent into a fiber at a repetition rate of say
1 GHz with pulse widths of about 100 ps (much shorter than TB), they can still be treated using
the quasi-cw analysis presented above because the time interval between successive “1” bits is
usually short enough that they can pump an acoustic wave coherently. On average, the only
effect is to increase the Brillouin threshold by a factor of approximately 2. Further increase in the
threshold can be achieved by phase modulating the carrier wave at 100 MHz (or faster) to
increase its spectral bandwidth.
Brillouin fiber lasers
The Brillouin gain can be used to make lasers by placing the fiber in a ring or mirrored
cavity. For a ring geometry, the boundary condition IS (L) = RIS (0) implies that Eq. (6)
becomes
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Rexp(g B PL / A) = 1 .
(22)
Consequently the factor of 21 in Eq. (7) is replaced with a value between 0.1 and 1 (depending
on R) if L is less than about 100 m. For short fibers, such that the longitudinal mode spacing
L = c / nL is larger than the Brillouin gain bandwidth B , the laser operates stably in a
single longitudinal mode.
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